• If an experiment consists of a single trial and the outcome of the trial can only be either a success
or a failure, then the trial is called a
Bernoulli trial. • The number of success X in one Bernoulli trial, which can be 1 or
0, is a Bernoulli random variable.
• Note: If p is the probability of success in a Bernoulli experiment,
the EX = p and VX = p1 – p.
The terms success and failure are simply statistical terms, and do not have positive or negative implications. In a production setting, finding a
defective product may be termed a “success,” although it is not a positive result.
3-3 Bernoulli Random Variable 3-3 Bernoulli Random Variable
Consider a Bernoulli Process
in which we have a sequence of n identical
trials satisfying the following conditions: 1. Each trial has two possible outcomes, called
success and
failure .
The two outcomes are mutually exclusive
and exhaustive
. 2. The
probability of success , denoted by
p , remains
constant from trial
to trial. The probability of failure
is denoted by q
, where q = 1-p
.
3. The n trials are independent
. That is, the outcome of any trial does not affect the outcomes of the other trials.
A random variable, X, that counts the number of successes in n Bernoulli trials, where p is the probability of success in any given trial, is said to
follow the binomial probability distribution with parameters n number of trials and p probability of success. We call X the binomial
random variable.
The terms success and failure are simply statistical terms, and do not have positive or negative implications. In a production setting, finding a defective product may be termed a “success,” although it is not a positive result.
3-4 The Binomial Random Variable 3-4 The Binomial Random Variable
Suppose we toss a single fair and balanced coin five times in succession, and let X represent the number of heads.
There are 2
5
= 32 possible sequences of H and T S and F in the sample space for this experiment. Of these, there are 10 in which there are exactly 2 heads X=2:
HHTTT HTHTH HTTHT HTTTH THHTT THTHT THTTH TTHHT TTHTH TTTHH
The probability of each of these 10 outcomes is p
3
q
3
= 12
3
12
2
=132, so the probability of 2 heads in 5 tosses of a fair and balanced coin is:
PX = 2 = 10 132 = 1032 = 0.3125
10 132
Number of outcomes with 2 heads
Probability of each outcome with 2 heads
Binomial Probabilities Introduction Binomial Probabilities Introduction
10 132
Number of outcomes with 2 heads
Probability of each outcome with 2 heads
PX=2 = 10 132 = 1032 = .3125
Notice that this probability has two parts:
In general: 1. The
probability of a given sequence
of x successes out of n trials with probability of success p and
probability of failure q is equal to:
p
x
q
n-x
nCx n
x n
x n x
2. The number of different sequences
of n trials that result in exactly x successes is equal to the number
of choices of x elements out of a total of n elements. This number is denoted:
Binomial Probabilities continued Binomial Probabilities continued
Number of successes, x Probability Px
1 2
3 n
1.00 n
n p q
n n
p q n
n p q
n n
p q
n n n
n p q
n n
n n
n n n
1 1
2 2
3 3
1 1
2 2
3 3
The binomial probability distribution
:
where : p is the probability of success in a single trial,
q = 1-p, n is the number of trials, and
x is the number of successes.
P x n
x p q
n x n x
p q
x n x
x n x
The Binomial Probability Distribution The Binomial Probability Distribution
n=5 p
x
0.01 0.05
